Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
Generated by claude-sonnet-4-6 · 2026-06-15 10:24 · grounding rag
Model Answer
Given: Two concentric circles with centre O, radii R = 5 cm and r = 3 cm. AB is a chord of the larger circle touching the smaller circle at P.
To find: Length of AB.
Since AB is a tangent to the smaller circle at P, by Theorem 10.1:
$$OP \perp AB$$
In right triangle OPB:
$$OB^2 = OP^2 + PB^2$$
$$5^2 = 3^2 + PB^2$$
$$PB^2 = 25 - 9 = 16$$
$$PB = 4 \text{ cm}$$
Since the perpendicular from the centre bisects the chord, AP = PB = 4 cm.
$$AB = 2 \times PB = 2 \times 4 = \textbf{8 cm}$$
Source: Chapter 10, Section 10.2 (Theorem 10.1) and Exercise 10.2 Q.7
Explanation
- The key property used: a tangent is perpendicular to the radius at the point of contact (Theorem 10.1), so OP ⊥ AB.
- Since OP ⊥ AB and O is the centre, OP bisects chord AB → AP = PB.
- Apply Pythagoras in △OPB with OB = 5 (radius of larger circle) and OP = 3 (radius of smaller circle).
- Examiners expect the diagram reasoning (perpendicularity + bisection) stated clearly before the calculation.